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Arakelov几何（Arakelov Geometry）（影印版）

样章
作者：
Atsushi Moriwaki
定价：
135.00元
版面字数：
500.00千字
开本：
16开
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2026-01-05
ISBN：
978-7-04-065929-0
物料号：
65929-00
出版时间：
2026-03-27
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
代数几何学
暂无
  • 前辅文
  • Preface
  • Chapter 1. Preliminaries
    • 1.1. Frequently used notation and conventions
    • 1.2. Normed finite-dimensional vector space
    • 1.3. Lemmas on the length of modules
    • 1.4. Image of a homomorphism and its determinant
    • 1.5. Norm of flat and finite homomorphisms
    • 1.6. Principal divisor and Weil’s reciprocity law
    • 1.7. Existence of rational sections not passing through given points
    • 1.8. Graded modules and ample invertible sheaves
    • 1.9. Several results on separable extensions of the base field
    • 1.10. Determinant bundle
    • 1.11. Complex manifold and Hodge theory
    • 1.12. Connection and curvature
    • 1.13. Poincaré-Lelong formula
    • 1.14. C∞ on reduced complex space
  • Chapter 2. Geometry of Numbers
    • 2.1. Convex set and Minkowski’s theorem
    • 2.2. Polar dual set and Mahler’s inequality
    • 2.3. The Brunn-Minkowski theorem
    • 2.4. Estimate of the number of points in a convex lattice
    • 2.5. Normed finitely generated Z-module
    • 2.6. λQ and λZ
  • Chapter 3. Arakelov Geometry on Arithmetic Curves
    • 3.1. Orders
    • 3.2. Arithmetic Chow group over a reduced order
    • 3.3. Hermitian R-module
    • 3.4. Arithmetic Riemann-Roch formula on arithmetic curves
    • 3.5. Effective estimate of the number of small sections
    • 3.6. Several formulae on arithmetic degree
    • 3.7. Volume exactness
    • 3.8. Ample invertible sheaves on arithmetic curves
  • Chapter 4. Arakelov Geometry on Arithmetic Surfaces
    • 4.1. Deligne’s pairing
    • 4.2. Green functions on Riemann surfaces
    • 4.3. Arithmetic Chow groups on arithmetic surfaces
    • 4.4. Intersection theory on arithmetic surfaces
    • 4.5. Arakelov metric of dualizing sheaf and adjunction formula
    • 4.6. Determinant bundles for curves
    • 4.7. Faltings’ Riemann-Roch theorem on arithmetic surfaces
    • 4.8. Determinant bundle and theta divisor
    • 4.9. Existence of Faltings’ metric
  • Chapter 5. Arakelov Geometry on General Arithmetic Varieties
    • 5.1. Preliminaries on algebraic geometry and complex geometry
    • 5.2. Intersection theory of Cartier divisors on excellent schemes
    • 5.3. Higher dimensional generalization of Weil’s reciprocity law in complex geometry
    • 5.4. Intersection theory on arithmetic varieties
    • 5.5. Characteristic classes and Bott-Chern secondary characteristic form
    • 5.6. Arithmetic characteristic classes
    • 5.7. Arithmetic Riemann-Roch formula
    • 5.8. Multi-indexed version of Gromov’s inequality
    • 5.9. Arithmetic Hilbert-Samuel formula
    • 5.10. Several kinds of positivity of C∞-hermitian invertible sheaves
    • 5.11. Estimation of λQ for a normed graded ring
  • Chapter 6. Arithmetic Volume Function and Its Continuity
    • 6.1. Arithmetic volume function
    • 6.2. Extension of volume function over Q
    • 6.3. Continuity of volume function
    • 6.4. Generalized Hodge index theorem
    • 6.5. Estimate of the number of small sections
  • Chapter 7. Nakai-Moishezon Criterion on an Arithmetic Variety
    • 7.1. Endmorphism N and its basic properties
    • 7.2. Bounded extension of holomorphic sections
    • 7.3. Proof of Nakai-Moishezon’s criterion on an arithmetic variety
    • 7.4. Arithmetic Hilbert-Samuel formula
  • Chapter 8. Arithmetic Bogomolov Inequality
    • 8.1. Semistable locally free coherent sheaves on algebraic curves
    • 8.2. Hermite-Einstein metric and stability
    • 8.3. Arithmetic Bogomolov inequality and its proof
  • Chapter 9. Lang-Bogomolov Conjecture
    • 9.1. Height function
    • 9.2. Height function on abelian variety
    • 9.3. Equidistribution theorem
    • 9.4. Cubic metric on complex abelian variety
    • 9.5. Bogomolov’s conjecture
    • 9.6. The Lang-Bogomolov Conjecture
    • 9.7. Small points with respect to a subgroup of finite rank
    • 9.8. The proof of Theorem 9.24
  • Bibliography
  • Index

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